These values of \(c\)  are further shown to be minimax or near-minimax optimal for cluster recovery. We refer to Theorems 3
and 4 in [1]} for the precise expressions and details. Under these minimal conditions on \(\lambda _{\min }\left(_{  }^*\right)\) , the exact recovery of the clusters holds with probability larger than \(1 - 1/(d \vee n)\) . This will allow us to show, in Section , that inference in cluster-based graphical models is not hampered by the clustering step.
